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Theorem notzfausOLD 5269
Description: Obsolete proof of notzfaus 5268 as of 18-Nov-2023. (Contributed by NM, 8-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
notzfaus.1 𝐴 = {∅}
notzfaus.2 (𝜑 ↔ ¬ 𝑥𝑦)
Assertion
Ref Expression
notzfausOLD ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem notzfausOLD
StepHypRef Expression
1 notzfaus.1 . . . . . 6 𝐴 = {∅}
2 0ex 5214 . . . . . . 7 ∅ ∈ V
32snnz 4706 . . . . . 6 {∅} ≠ ∅
41, 3eqnetri 3012 . . . . 5 𝐴 ≠ ∅
5 n0 4275 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
64, 5mpbi 233 . . . 4 𝑥 𝑥𝐴
7 biimt 364 . . . . . 6 (𝑥𝐴 → (𝑥𝑦 ↔ (𝑥𝐴𝑥𝑦)))
8 iman 405 . . . . . . 7 ((𝑥𝐴𝑥𝑦) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥𝑦))
9 notzfaus.2 . . . . . . . 8 (𝜑 ↔ ¬ 𝑥𝑦)
109anbi2i 626 . . . . . . 7 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝑦))
118, 10xchbinxr 338 . . . . . 6 ((𝑥𝐴𝑥𝑦) ↔ ¬ (𝑥𝐴𝜑))
127, 11bitrdi 290 . . . . 5 (𝑥𝐴 → (𝑥𝑦 ↔ ¬ (𝑥𝐴𝜑)))
13 xor3 387 . . . . 5 (¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ (𝑥𝑦 ↔ ¬ (𝑥𝐴𝜑)))
1412, 13sylibr 237 . . . 4 (𝑥𝐴 → ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)))
156, 14eximii 1844 . . 3 𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑))
16 exnal 1834 . . 3 (∃𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
1715, 16mpbi 233 . 2 ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
1817nex 1808 1 ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wex 1787  wcel 2111  wne 2941  c0 4251  {csn 4555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-nul 5213
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2942  df-v 3422  df-dif 3883  df-nul 4252  df-sn 4556
This theorem is referenced by: (None)
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